Probabilistic Guarantees for Abductive Inference
Abstract: Abductive reasoning is ubiquitous in artificial intelligence and everyday thinking. However, formal theories that provide probabilistic guarantees for abductive inference are lacking. We present a quantitative formalization of abductive logic that combines Bayesian probability with the interpretation of abduction as a search process within the Algorithmic Search Framework (ASF). By incorporating uncertainty in background knowledge, we establish two novel sets of probabilistic bounds on the success of abduction when (1) selecting the single most likely cause while assuming noiseless observations, and (2) selecting any cause above some probability threshold while accounting for noisy observations. To our knowledge, no existing abductive or general inference bounds account for noisy observations. Furthermore, while most existing abductive frameworks assume exact underlying prior and likelihood distributions, we assume only percentile-based confidence intervals for such values. These milder assumptions result in greater flexibility and applicability of our framework. We also explore additional information-theoretic results from the ASF and provide mathematical justifications for everyday abductive intuitions.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: We have added revisions to address the majority of reviewer feedback. Detailed descriptions of each change are included in official comments to each of the reviewers. A summary of the changes is as follows:
- Added discussion of future extensions to creative abduction (section 7)
- Added explanation of continuous/categorical-based observations (section 3.1)
- Clarified that the framework accounts for multiple concurrent causes so long as the selected causes are disjoint; corrected unclear language (section 3.1)
- Condensed earlier sections of the paper (mainly sections 2 and 3)
- Corrected typo discussing MLE, improved explanation for analogizing MAP to abduction (section 2.2)
- Inclusion/a brief discussion of do-calculus in section 3.2
- Added discussion justifying the use and specific advantages of the ASF, in addition to novel results produced (start of section 5)
- Added additional description for external information resource $F$ (section 5.1)
- Added additional sources to section 2.3
- Fixed missing references in sections 3 and 4
- Provided additional clarification about likelihood functions (section 3.2)
- Provided additional explanation of Bayesian neural networks for estimating posterior distributions (paragraph 3 of section 6)
- Corrected typo that the sum of components in simplex vector $\mathbf{c} \in \mathcal{S}$ is one (not less than or equal to 1)
New changes (7/20/2024):
- Added an appendix section "Toy Example and Additional Derivations of Theorem 4.2". This section goes through a toy example extending Table 2 following steps in section 4 with intuitive explanations, and derives final general bounds of abductive success.
- Added derivations of two new directly computable forms of Theorem 4.2 with different setup constraints.
- Uploaded code for experiment
- Revised notation of new appendix section, added results of bound derivations for every observation vector and cause
Assigned Action Editor: ~Manuel_Haussmann1
Submission Number: 2737
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